Benjamin–Bona–Mahony equation
The Benjamin–Bona–Mahony equation (or BBM equation) is the partial differential equation
\[u_t+u_x+uu_x-u_{xxt}=0.\,\]
This equation was introduced in Benjamin, Bona, and Mahony (1972) as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long waves of small amplitude in 1+1 dimensions. They show the stability and uniqueness of solutions to the BBM equation. This contrasts with the KdV equation, which is unstable in its high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three.[1]
A higher-dimensional version is given by
\[u_t-\nabla^2u_t+\operatorname{div}\,\varphi(u)=0.\,\]
where \(\varphi\) is a fixed smooth function from \(\mathbb R\) to \(\mathbb R^n\). Avrin & Goldstein (1985) proved local existence of a solution in all dimensions.
See also
Notes
References
- Avrin, Joel; Goldstein, Jerome A. (1985), "Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions.", Nonlinear Anal. 9 (8): 861–865, doi:10.1016/0362-546X(85)90023-9, MR 0799889
- Benjamin, T. B.; Bona, J. L.; Mahony, J. J. (1972), "Model Equations for Long Waves in Nonlinear Dispersive Systems", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 272 (1220): 47–78, Bibcode 1972RSPTA.272...47B, doi:10.1098/rsta.1972.0032, ISSN 0962-8428, JSTOR 74079
- Zwillinger, Daniel (1998), Handbook of differential equations (3rd ed.), Boston, MA: Academic Press, Inc., pp. 174, 176, ISBN 978-0-12-784396-4, MR 0977062 (Warning: On p. 174 Zwillinger misstates the Benjamin–Bona–Mahony equation, confusing it with the similar KdV equation.)
